The 18-year-old picked out the top corner from distance in the 71st minute as a much-changed Reds side managed to secure their sixth straight victory in all competitions.
Milner limped off the field moments later after picking up an issue in a challenge with Walcott, with Yasser Larouci introduced for his debut as Liverpool's side became even more inexperienced.
Adam Lallana then sent a long-range effort over the crossbar in the 20th minute during a period where Liverpool were starting to enjoy more of the ball in Everton's half of the field.
Neco Williams was next to try his luck for Klopp's side as he had an effort deflected wide in the 26th minute, before Adrian did excellently once again to keep out a strike from Richarlison down the other end.
The visitors had another half-chance just past the 30-minute mark when Walcott delivered a dangerous cross towards Calvert-Lewin, but the in-form attacker could not make proper contact on the stretch.
Jones then struck wide of the Everton goal in the 33rd minute before Minamino headed a Divock Origi cross just wide of the far post moments later after finding space inside the penalty box.
Calvert-Lewin fired wide of the Liverpool post late in the half with Ancelotti's side searching for the opener, but the standout moment in the first 45 minutes occurred moments later when Jordan Pickford got down to keep out a strike from Origi which looked to be heading for the bottom corner.
Schneiderlin had the first half-chance of the second period after a smart piece of work from Djibril Sidibe, but the midfielder could only curl over the crossbar from distance.
Liverpool's first opportunity after the restart fell to Williams, with the teenager testing the reactions of Pickford from distance during a spell where the home side were on the front foot.
Indeed, the European Cup holders were the team forcing the issue as the game passed the hour, although there was still no sign of the opening goal in the latest instalment of the Merseyside derby.
Not too much changed in the immediate aftermath, though, with Liverpool still looking the more likely to make the breakthrough as the contest headed for its final 20 minutes.
Pickford had to keep out a long-range strike from Origi in the 70th minute before Minamino's debut came to an end as Alex Oxlade-Chamberlain entered the field.
The home side finally made the breakthrough in the 71st minute when Jones picked out the far corner with a wonderful effort from distance, sparking huge celebrations inside the stadium.
Pickford then had to keep out a low strike from Oxlade-Chamberlain with Liverpool threatening a second. Everton needed a response, but the visitors were finding it increasingly difficult to harm their rivals, who were comfortable heading into the final five minutes of action.
The Toffees were given three additional minutes at the end of the 90 to force a replay but in the end it was comfortable for Klopp's team, who eased their way into the fourth round of the famous competition.
Liverpool are back in Premier League action away to Jose Mourinho's Tottenham Hotspur on Saturday, while Everton will host Brighton & Hove Albion on the same afternoon.
LIVERPOOL (4-2-3-1): Adrian; Milner (Larouci 9'), Phillips, Gomez, Williams; Lallana, Chirivella; Jones, Minamino (Oxlade-Chamberlain 70'), Elliott (Brewster 79'); Origi
EVERTON (3-4-3): Pickford; Coleman (Kean 63'), Holgate, Mina, Digne; Sidibe, Sigurdsson (Delph 63'), Schneiderlin, Walcott (Bernard 79'); Richarlison, Calvert-Lewin